1. Correlation and regression
Introduction to Statistical Methods for Measuring “Omics” and Field Data
Correlation and regression

2. Overview
Correlation
Simple Linear Regression

3. Correlation

4. General Overview of Correlational Analysis
The purpose is to measure the strength of a linear relationship between 2 variables.
A correlation coefficient does not ensure “causation” (i.e. a change in X causes a change in Y)
X is typically the input, measured, or independent variable.
Y is typically the output, predicted, or dependent variable.
If X increases and there is a predictable shift in the values of Y, a correlation exists.

5. General Properties of Correlation Coefficients
Values can range between +1 and -1
The value of the correlation coefficient represents the scatter of points on a scatterplot
You should be able to look at a scatterplot and estimate what the correlation would be
You should be able to look at a correlation coefficient and visualize the scatterplot

6. Interpretation
Depends on what the purpose of the study is… but here is a “general guideline”...
Value = magnitude of the relationship
Sign = direction of the relationship

7. Correlation graph Strong relationships Weak relationships Y Y
Positive correlation X X Y Y
Negtaive correlation X X

8. The Pearson Correlation Coefficient

9. Correlation Coefficient
The correlation coefficient is a measure of the strength and the direction of a linear relationship between two variables. The symbol r represents the sample correlation coefficient. The formula for r is
The range of the correlation coefficient is 1 to 1. If x and y have a strong positive linear correlation, r is close to 1. If x and y have a strong negative linear correlation, r is close to 1. If there is no linear correlation or a weak linear correlation, r is close to 0.

10. Calculating a Correlation Coefficient
In Words In Symbols
Find the sum of the x-values.
Find the sum of the y-values.
Multiply each x-value by its corresponding y-value and find the sum.
Square each x-value and find the sum.
Square each y-value and find the sum.
Use these five sums to calculate the correlation coefficient.
Continued.

11. Correlation Coefficient
Example:
Calculate the correlation coefficient r for the following data. x y 1
– 3 2
– 1 3 4 5

12. Correlation Coefficient
Example:
Calculate the correlation coefficient r for the following data. x y xy x2 y2 1
– 3 9 2
– 1
– 2 4 3 16 5 10 25

13. Correlation Coefficient
Example:
Calculate the correlation coefficient r for the following data. x y xy x2 y2 1
– 3 9 2
– 1
– 2 4 3 16 5 10 25
There is a strong positive linear correlation between x and y.

14. Significance Test for Correlation
Hypotheses
H0: ρ = 0 (no correlation)
HA: ρ ≠ 0 (correlation exists)
Test statistic
(with n – 2 degrees of freedom)

15. Linear Regression

16. Linear regression
Deals with relationship between two variables X and Y.
Y is the variables whose “behavior” we wish to study ( e.g., fuel efficiency in a car).
X is the variable we believe would help explain the behavior of Y (e.g., the size of the car).

17. Regression model
The simple linear regression model:

18. Components of the models

19. Regression Line
A regression line, also called a line of best fit, is the line for which the sum of the squares of the residuals is a minimum.
The Equation of a Regression Line
The equation of a regression line for an independent variable x and a dependent variable y is
ŷ = mx + b
where ŷ is the predicted y-value for a given x-value. The slope m and y-intercept b are given by

20. Regression Line x y 1 – 3 2 – 1 3 4 5 Example:
Find the equation of the regression line. x y 1
– 3 2
– 1 3 4 5
Continued.

21. Regression Line x y xy x2 y2 1 – 3 9 2 – 1 – 2 4 3 16 5 10 25 Example:
Find the equation of the regression line. x y xy x2 y2 1
– 3 9 2
– 1
– 2 4 3 16 5 10 25
Continued.

22. Regression Line x y xy x2 y2 1 – 3 9 2 – 1 – 2 4 3 16 5 10 25 Example:
Find the equation of the regression line. x y xy x2 y2 1
– 3 9 2
– 1
– 2 4 3 16 5 10 25
Continued.

23. Regression Line x y xy x2 y2 1 – 3 9 2 – 1 – 2 4 3 16 5 10 25 Example:
Find the equation of the regression line. x y xy x2 y2 1
– 3 9 2
– 1
– 2 4 3 16 5 10 25
Continued.

24. Regression Line Hours, x Test score, y xy x2 y2 1 2 3 5 6 7 10 96 85
Example:
The following data represents the number of hours 12 different students watched television during the weekend and the scores of each student who took a test the following Monday.
a.) Find the equation of the regression line.
b.) Use the equation to find the expected test score for a student who watches 9 hours of TV.
Hours, x 1 2 3 5 6 7 10
Test score, y 96 85 82 74 95 68 76 84 58 65 75 50 xy
164
222
285
340
380
420
348
455
525
500 x2 4 9 25 36 49
100 y2
9216
7225
6724
5476
9025
4624
5776
7056
3364
4225
5625
2500

25. Regression Line Continued. Example continued: 100 x y
Hours watching TV
Test score 80 60 40 20 2 4 6 8 10
ŷ = –4.07x
Continued.

26. Regression Line Example continued:
Using the equation ŷ = -4.07x , we can predict the test score for a student who watches 9 hours of TV.
ŷ = –4.07x
= –4.07(9)
= 57.34
A student who watches 9 hours of TV over the weekend can expect to receive about a on Monday’s test.

27. Variation About a Regression Line
The total variation about a regression line is the sum of the squares of the differences between the y-value of each ordered pair and the mean of y.
The explained variation is the sum of the squares of the differences between each predicted y-value and the mean of y.
The unexplained variation is the sum of the squares of the differences between the y-value of each ordered pair and each corresponding predicted y-value.

28. Coefficient of Determination
The coefficient of determination R2 is the ratio of the explained variation to the total variation. That is,
Example:
The correlation coefficient for the data that represents the number of hours students watched television and the test scores of each student is r   Find the coefficient of determination.
About 69.1% of the variation in the test scores can be explained by the variation in the hours of TV watched. About 30.9% of the variation is unexplained.

29. Regression hypothesis

33. RStudio
Function cor.test is used to calculate correlation r, and t statistics.
Function lm is used to calculate regression
Example:
Hours, x 1 2 3 5 6 7 10
Test score, y 96 85 82 74 95 68 76 84 58 65 75 50
X<-c(0,1,2,3,3,5,5,5,6,7,7,10)
Y<-c(96,85,82,74,95,68,76,84,58,65,75,50)
cor.test(X,Y)
G<-lm(X~Y)
Summary(G)

34. RStudio Count<-c(9,25,15,2,14,25,24,47) > Count
[1]
Speed<-c(2,3,5,9,14,24,29,34)
G<-lm(Count~Speed)
> summary(G)
Call:
lm(formula = Count ~ Speed)
Residuals:
Min Q Median Q Max
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept)
Speed *
---
Signif. codes: 0 ‘***’ ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: on 6 degrees of freedom
Multiple R-squared: , Adjusted R-squared:
F-statistic: on 1 and 6 DF, p-value: